sbv-8.6: SMT Based Verification: Symbolic Haskell theorem prover using SMT solving.

Documentation.SBV.Examples.WeakestPreconditions.GCD

Description

Proof of correctness of an imperative GCD (greatest-common divisor) algorithm, using weakest preconditions. The termination measure here illustrates the use of lexicographic ordering. Also, since symbolic version of GCD is not symbolically terminating, this is another example of using uninterpreted functions and axioms as one writes specifications for WP proofs.

Synopsis

# Program state

data GCDS a Source #

The state for the sum program, parameterized over a base type a.

Constructors

 GCDS Fieldsx :: aFirst valuey :: aSecond valuei :: aCopy of x to be modifiedj :: aCopy of y to be modified
Instances
 Source # Instance details Methodsfmap :: (a -> b) -> GCDS a -> GCDS b #(<$) :: a -> GCDS b -> GCDS a # Source # Instance details Methodsfold :: Monoid m => GCDS m -> m #foldMap :: Monoid m => (a -> m) -> GCDS a -> m #foldr :: (a -> b -> b) -> b -> GCDS a -> b #foldr' :: (a -> b -> b) -> b -> GCDS a -> b #foldl :: (b -> a -> b) -> b -> GCDS a -> b #foldl' :: (b -> a -> b) -> b -> GCDS a -> b #foldr1 :: (a -> a -> a) -> GCDS a -> a #foldl1 :: (a -> a -> a) -> GCDS a -> a #toList :: GCDS a -> [a] #null :: GCDS a -> Bool #length :: GCDS a -> Int #elem :: Eq a => a -> GCDS a -> Bool #maximum :: Ord a => GCDS a -> a #minimum :: Ord a => GCDS a -> a #sum :: Num a => GCDS a -> a #product :: Num a => GCDS a -> a # Source # Instance details Methodstraverse :: Applicative f => (a -> f b) -> GCDS a -> f (GCDS b) #sequenceA :: Applicative f => GCDS (f a) -> f (GCDS a) #mapM :: Monad m => (a -> m b) -> GCDS a -> m (GCDS b) #sequence :: Monad m => GCDS (m a) -> m (GCDS a) # (SymVal a, SMTValue a) => Fresh IO (GCDS (SBV a)) Source # Fresh instance for the program state Instance details Methodsfresh :: QueryT IO (GCDS (SBV a)) Source # Show a => Show (GCDS a) Source # Instance details MethodsshowsPrec :: Int -> GCDS a -> ShowS #show :: GCDS a -> String #showList :: [GCDS a] -> ShowS # (SymVal a, Show a) => Show (GCDS (SBV a)) Source # Show instance for GCDS. The above deriving clause would work just as well, but we want it to be a little prettier here, and hence the OVERLAPS directive. Instance details MethodsshowsPrec :: Int -> GCDS (SBV a) -> ShowS #show :: GCDS (SBV a) -> String #showList :: [GCDS (SBV a)] -> ShowS # Generic (GCDS a) Source # Instance details Associated Typestype Rep (GCDS a) :: Type -> Type # Methodsfrom :: GCDS a -> Rep (GCDS a) x #to :: Rep (GCDS a) x -> GCDS a # Mergeable a => Mergeable (GCDS a) Source # Instance details MethodssymbolicMerge :: Bool -> SBool -> GCDS a -> GCDS a -> GCDS a Source #select :: (Ord b, SymVal b, Num b) => [GCDS a] -> GCDS a -> SBV b -> GCDS a Source # type Rep (GCDS a) Source # Instance details type Rep (GCDS a) = D1 (MetaData "GCDS" "Documentation.SBV.Examples.WeakestPreconditions.GCD" "sbv-8.6-DBvpUlQTqMsBJjjQk7wU8u" False) (C1 (MetaCons "GCDS" PrefixI True) ((S1 (MetaSel (Just "x") NoSourceUnpackedness NoSourceStrictness DecidedLazy) (Rec0 a) :*: S1 (MetaSel (Just "y") NoSourceUnpackedness NoSourceStrictness DecidedLazy) (Rec0 a)) :*: (S1 (MetaSel (Just "i") NoSourceUnpackedness NoSourceStrictness DecidedLazy) (Rec0 a) :*: S1 (MetaSel (Just "j") NoSourceUnpackedness NoSourceStrictness DecidedLazy) (Rec0 a)))) type G = GCDS SInteger Source # Helper type synonym # The algorithm The imperative GCD algorithm, assuming strictly positive x and y:  i = x j = y while i != j -- While not equal if i > j i = i - j -- i is greater; reduce it by j else j = j - i -- j is greater; reduce it by i  When the loop terminates, i equals j and contains GCD(x, y). Symbolic GCD as our specification. Note that we cannot really implement the GCD function since it is not symbolically terminating. So, we instead uninterpret and axiomatize it below. NB. The concrete part of the definition is only used in calls to traceExecution and is not needed for the proof. If you don't need to call traceExecution, you can simply ignore that part and directly uninterpret. In that case, we simply use Prelude's version. Constraints and axioms we need to state explicitly to tell the SMT solver about our specification for GCD. pre :: G -> SBool Source # Precondition for our program: x and y must be strictly positive Postcondition for our program: i == j and i = gcd x y Stability condition: Program must leave x and y unchanged. A program is the algorithm, together with its pre- and post-conditions. # Correctness With the axioms in place, it is trivial to establish correctness: >>> correctness Total correctness is established. Q.E.D.  Note that I found this proof to be quite fragile: If you do not get the algorithm right or the axioms aren't in place, z3 simply goes to an infinite loop, instead of providing counter-examples. Of course, this is to be expected with the quantifiers present. # Concrete execution Example concrete run. As we mentioned in the definition for gcd, the concrete-execution function cannot deal with uninterpreted functions and axioms for obvious reasons. In those cases we revert to the concrete definition. Here's an example run: >>> traceExecution imperativeGCD$ GCDS {x = 14, y = 4, i = 0, j = 0}
*** Precondition holds, starting execution:
{x = 14, y = 4, i = 0, j = 0}
===> [1.1] Conditional, taking the "then" branch
{x = 14, y = 4, i = 0, j = 0}
===> [1.1.1] Skip
{x = 14, y = 4, i = 0, j = 0}
===> [1.2] Assign
{x = 14, y = 4, i = 14, j = 4}
===> [1.3] Loop "i != j": condition holds, executing the body
{x = 14, y = 4, i = 14, j = 4}
===> [1.3.{1}] Conditional, taking the "then" branch
{x = 14, y = 4, i = 14, j = 4}
===> [1.3.{1}.1] Assign
{x = 14, y = 4, i = 10, j = 4}
===> [1.3] Loop "i != j": condition holds, executing the body
{x = 14, y = 4, i = 10, j = 4}
===> [1.3.{2}] Conditional, taking the "then" branch
{x = 14, y = 4, i = 10, j = 4}
===> [1.3.{2}.1] Assign
{x = 14, y = 4, i = 6, j = 4}
===> [1.3] Loop "i != j": condition holds, executing the body
{x = 14, y = 4, i = 6, j = 4}
===> [1.3.{3}] Conditional, taking the "then" branch
{x = 14, y = 4, i = 6, j = 4}
===> [1.3.{3}.1] Assign
{x = 14, y = 4, i = 2, j = 4}
===> [1.3] Loop "i != j": condition holds, executing the body
{x = 14, y = 4, i = 2, j = 4}
===> [1.3.{4}] Conditional, taking the "else" branch
{x = 14, y = 4, i = 2, j = 4}
===> [1.3.{4}.2] Assign
{x = 14, y = 4, i = 2, j = 2}
===> [1.3] Loop "i != j": condition fails, terminating
{x = 14, y = 4, i = 2, j = 2}
*** Program successfully terminated, post condition holds of the final state:
{x = 14, y = 4, i = 2, j = 2}
Program terminated successfully. Final state:
{x = 14, y = 4, i = 2, j = 2}


As expected, gcd 14 4 is 2.